There is a book that not many physicists I know of seem to like (except mathematical physicists, of course), but that is a true gem in the eyes of mathematicians: I am referring to V. Arnold Mathematical Methods of Classical Mechanics, here on amazon.
In this book, which is in the short list (number 12, to be precise) of my fundamental math book across all math fields, Chapter VIII is entirely devoted to differential forms.
If you read it, you have, I believe, an excellent answer.
One small suggestion to build understanding: DISCRETIZE. Do not think of fancy integrals, simply think that 0-forms are scalars, 1-forms oriented segments, 2-forms oriented areas, and that integration over them is simply sums. Now "prove" Stokes theorem for simple tiny cubes, and notice that the definition of the derivatives of forms is exactly done to keep track of faces. At the infinitesimal level, it is just book keeping.
If I ever had to teach a basic class on forms, I would do precisely that: discretize first.